I think your expectations of your son are too high given all he is going through.
Three year olds need stability and familiarity. He can’t absorb the trauma of your sudden new living arrangements being forced on him for half the week every week. It’s not only confusing, it’s emotionally disruptive with far reaching consequences. Some children manage it better than others.
Having to move house twice a week is too much for him and nobody should blame him for complaining about that.
Children get a bad deal when expected to live like that, especially without warning or preparation. (And presumably he is expected to do it, relentlessly for the rest of his childhood)
You may have wanted to move to a completely new life with a new house and a new partner.
But your little boy didn’t.
He just wants to stay home ..the home he feels emotionally connected to, with the same two parents he’s had all his life.
It’s rather sad that his mum doesn’t appear to understand how impactful this change has been and will continue to be for him. It’s a massive emotional burden for him to have to carry.
How would you feel OP, if you were forced to move between two houses every 3 or 4 days, relentlessly, against your will, for the next 10 years.
Might you complain a little bit? Might it affect you psychologically?
Maybe you do actually understand all the implications and are just looking for advice to lessen the burden on you of him crying and being upset about it.
The only advice that can be given (other than special,treats like letting him choose his own new bedroom etc) is that your little boy should be your absolute priority, with patience, love and understanding in abundance from you and his dad. You need to give him your full, undisturbed time for play and activities together. He needs to feel that he has your full attention at all times. Ensure that your new relationship never interferes with or fights for space with your time with your son. You must be a dedicated parent to him.
Good luck !